\begin{conjecture} Every planar graph without 4-cycles and 5-cycles is 3-colourable. \end{conjecture}

Borodin et al. [BGRS] who proved that every planar graph without cycles of length in $\{4, \dots ,7\}$ is 3-colourable.

Borodin and Raspaud [BR] proposed the following conjecture which implies Steinberg's Conjecture.

\begin{conjecture}[Strong Bordeaux Conjecture] Every planar graph without 5-cycles and without adjacent triangles is 3-colourable. \end{conjecture}

This conjecture is in turn implied by the following stronger one due to Borodin et al. [BGJR] \begin{conjecture} Every planar graph without 3-cycles adjacent to cycles of length 3 or 5 is 3-colourable. \end{conjecture} Borodin et al. [BGJR06] proved that every planar graph without triangles adjacent to cycles of length from 3 to 9 is 3-colourable.

Steinberg's Conjecture cannot been generalized to list colouring: Voigt [V] constructed a planar graph without 4-cycles and 5-cycles which is not 3-choosable.

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